"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

The puzzle of Galois descent

Suppose we have a system of polynomial equations over a perfect (to keep things simple) field , and we’d like to consider solutions of it over various field extensions of . Write for the set of all solutions to this system over .

As it happens, knowing for any algebraic extension of is equivalent to knowing , where denotes the algebraic closure of , together with the action of the absolute Galois group . After picking an embedding of into , the infinite Galois correspondence says that is precisely the set of fixed points of the closed subgroup of which stabilizes , and it’s not hard to see that this extends to ; that is, naturally acts on , and we have a natural identification

.

Now let’s categorify this situation. Before we considered, for each algebraic extension of , a set . There are many situations in mathematics in which it’s natural to consider instead a category, such that a morphism induces a functor , and so forth. The basic example is the case that is the category of -vector spaces, and for a morphism the corresponding functor is given by extension of scalars

.

This leads to many other examples coming from equipping vector spaces with extra structure: for example, might be

the category of representations of some finite group over ,

the category of commutative (or associative, or Lie) algebras over , or

the category of schemes over .

It would be great if understanding all of these categories was in some sense as simple as understanding the category , which generally tends to be simpler, and the action of the absolute Galois group on it, whatever that means. For example, the representation theory of finite groups over algebraically closed fields of characteristic zero is well understood, as is, say, the classification of semisimple Lie algebras. The general problem of trying to extract an understanding of from an understanding of is the problem of Galois descent.

We might very optimistically hope that the story here is directly analogous to the story above. This suggests the following puzzle.

Puzzle: In what sense could the statement be true for the examples given above?